You're a math anti-realist. *shrug*

You're a math anti-realist. — frank

Well, yes, but see the intro to my realism thread:

I wonder also if Anscombe's direction of fit works here. It's the difference between the list you take with you to remind yourself of what you want to buy and the list the register produces listing the things you actually purchased. The intent of the first list is to collect the things listed; of the second, to list the things collected. The first seeks to make the world fit the list, the second, to make the list to fit the world. Is it that anti-realism applies to ethics and aesthetics because we seek to make the world as we say, while realism applies to ontology and epistemology because we seek to make what we say fit the world? , — Banno

It's more a question of which, when than of either/or.

Are there five spoons on the table when we haven't yet counted them? That seems to be what we do. We might equally say that we don't know how many there are until we count them, but that is about what we beleive, not what is true.

If the math involved in structural engineering has worked for a hundred plus years, it seems very implausible to think it would suddenly cease working because we had found some contradictions in it that we previously had not known were there.

Because if number is real but not material, then you have something real but not material, meaning materialism is false. And that is a no-go in secular scientific culture. Ought not to over-complicate it. — Wayfarer

Of course, it doesn't matter because both materialism and idealism are metaphysical positions and, therefore, are neither true nor false, but, rather, useful or not. I read that mathematicians tend to be idealists and scientists tend to be materialists, which, given that, makes sense.

The intent of the first list is to collect the things listed; of the second, to list the things collected. The first seeks to make the world fit the list, the second, to make the list to fit the world — Banno

For the sake of simplicity, let's call them active and passive intentions.

With aesthetics, it's both. The beauty of the golden ratio and the perfect fifth were passively discovered, not created. But having discovered them, I may actively reproduce them. It's the same with ethics.

Though Pythagoras and his followers were literally violently opposed to irrational numbers, their appearance all around us couldn't ultimately be suppressed. So if you see the active in math, the passive also appears to be there.

As for the role consciousness plays in the production of the world we know, it's hard to deny some place for it. It's just a matter of how much, I suppose. Again, it's probably both.

For the sake of simplicity, let's call them active and passive intentions. — frank

They're already called world-to-word and word-to-world. Active and passive would confuse the issue.

But yes, that's the hypothesis of the other thread. Your post might be better there.

I wouldn't call it pointless to point at one consequence among many. — Olivier5

It's one consequence not just among many but among all. The supposed connection with the LEM does not hold.

I think he asked: Did we invent or did we discover chess? This I read as a parallel to the question of whether math are invented or discovered. — Olivier5

Thanks, @Olivier5

Within the context of a given mathematical system, yes. But there is more than one system, and hence more than one way to define/describe a line. For example, in analytical geometry a line is a collection of points, because that's just how analytical geometry is built up. — SophistiCat

Funny you should mention chess, because chess pieces are a good example of use-definition. A formal description of a chess game would not have a formal definition of a chess piece - it's just an abstract object to which we give a name. Its meaning is given by the use to which it is put in the game: the rules of how different pieces move, etc. — SophistiCat

Let's have consistency at least. Okay, more than one way to describe a line you say. Yet, you dismiss your own statement of "it's just an abstract object to which we give a name" regarding chess. So which is it? Chess exists in a vacuum. A line does not.

First, it only takes two points to make a line. Of course you can put however many points you want -- a collection of points. But we imply distance here, which doesn't change its meaning.

In short, I don't get your point.

If the math involved in structural engineering has worked for a hundred plus years, it seems very implausible to think it would suddenly cease working — Janus

The math involved in structural engineering have changed overtime. If in one of these changes, them engineers postulated that anything mathematical is both true and false at the same time, as Wittgenstein was effectively (though unwittingly) suggesting, they might have ended building quite a few failed bridges.

You know, maybe with a little good will you would be able to understand what I am saying. It's not that complicated. Otherwise, have fun with the other serial misunderstanders.

I don't get what it is you don't get, but let me address this bit:

Chess exists in a vacuum. A line does not. — Caldwell

No, chess does not exist in a vacuum, any more than a line. I think when people talk about Wittgenstein's "language games," and how math is "made up" because it is just a game we play (@Banno), they may be led astray by an association of the word "game" with something arbitrary and frivolous. But that's not at all true about literal games, such as chess, is it? If you make up an arbitrary game, it's going to be shit and no one will want to play it. And yet chess has been played for many centuries (and has evolved quite a bit over time). That doesn't just happen arbitrarily.

And the same is true about math, of course.

But you have misunderstood my position — Banno

I’m not saying that you yourself advocate philosophical materialism but that the general motivation for anti-realism in mathematics is that sans some notion of an incorporeal intelligence - which in pre-modern philosophy was assumed to be 'the divine intelligence' - the idea that numbers can be real outside of the human mind doesn’t make any sense. There’s no conceptual space for it. If number arises from counting, and if counting is something done by humans, then indeed maths is invented not discovered and it must be understood accordingly - which in practice means understanding how such an ability might have evolved.

Whereas the reaiist view is that integers (at least) are 'real in all possible worlds', and that humans simply evolved to the point of being able to recognise them. But that is not at all in keeping with the mainstream attitude which is generally nominalist and empiricist.

I've quoted a few times from a current article in Smithsonian magazine about 'What is Math' the following passage:

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

I think that really encapsulates the philosophical situation. It makes empiricists nervous, and empiricists hold sway.

(I've found that that very scholar, Robert James Brown, has written what seems a pretty authoritative (and extremely expensive!) text book on this subject, Platonism, Naturalism, and Mathematical Knowledge.)

I think that really encapsulates the philosophical situation. It makes empiricists nervous, and empiricists hold sway. — Wayfarer

I consider myself an empiricist, and yet I accept the existence of concepts. Am I doing something wrong?

I consider myself an empiricist, and yet I accept the existence of concepts. Am I doing something wrong? — Olivier5

I'm not accusing anyone of anything. The question is, what kind of existence conceptual information has. That is what is at issue. I think that brief passage I quoted expresses a pretty widely-held view. Am I believing something wrong?

Am I believing something wrong? — Wayfarer

No offence, but I think you pay too much attention to naïve materialists. You know, the kind of people who think their selves (i.e. themselves) don't exist because there can be no ghost in no machine... It seems to me that you resent their academic influence and credibility, but the world in which they are credible is a very small one and most probably not the world you live in. They only "hold sway" in a few academic circles that are irrelevant to anything.

I bet most people around you actually think of themselves as more than just meat puppets. And even the most naïve materialist out there will usually behave as a normal person, not as a meat puppet. They would for instance expect some respect from other human beings and would also extend some respect to other human beings, quite unlike the way they would treat actual puppets, and unlike the way they treat animals.

There's no reason to feel angry about academic fads. Academics only have the prestige that you give them.

maybe with a little good will you would be able to understand what I am saying. — Olivier5

I'm informing you that it is a basic misconception to think that the LEM is a consideration in the way you have claimed. That is not ill will.

Thanks for the info. Anything else you want to share?

I must congratulate you for learning, at long last, to use the quote feature. Well done!

Godel's 2nd incompleteness theorem is not that certain systems can't be proven consistent, but rather that if they are consistent then they can't be proved consistent by certain means — TonesInDeepFreeze

Constructively, the implications of the incompleteness theorems are stronger than that. The consistency of certain systems (PA and the like) cannot be constructively proved by any means. All that can potentially exist in the constructive sense with regards to such systems are i) potential constructive proofs of absolute inconsistency via brute-force evaluation of the underlying sequent calculus until the inconsistency is unearthed, and ii) conditional proofs of relative inconsistency, where the inconsistency of one system implies the inconsistency of another. E.g Gentzen proved that the inconsistency of PA implies the inconsistency of PRA + transfinite induction on the ordinals.

Personally, I am tempted to interpret Gentzen's result as denying the meaningfulness of epsilon zero (i.e the first limiting ordinal) as being considered a "well-founded" ordinal. For while it might be shown one day that PA is inconsistent, it can never be shown to be consistent unless one begs the question. By semi-decidability it is meaningful to ask if epsilon zero isn't well-founded, but it isn't meaningful to assume or hypothesise that it is well-founded,

If number arises from counting, and if counting is something done by humans, then indeed maths is invented not discovered and it must be understood accordingly - which in practice means understanding how such an ability might have evolved. — Wayfarer

As an aside, I've read of some scientific papers recently that indicate that children have at least a preliminary understanding of number from a very young age. This leads to the hypothesis that a sense of number is inborn, instinctual, just as our ability to learn and use language is.

I am well versed in the quote feature, as seen in my many posts in other threads. But I had been experimenting with not using it lately in order to avoid too much of a personal tit-for-tat kind of conversation. That is, to make my remarks general toward the points no matter who might have asserted them, though I recognize the disadvantages of not using the quote feature. Then, at a point, it became too cumbersome to reply without quoting, and then my experiment manifestly failed when you resorted to the personal claim that I lack "good will".

It's not that I bar myself from making personal remarks - indeed, I may liberally say quite what's on my mind about a poster's qualities - it's just that lately I was in the mood for experimenting with ways that might avoid my receiving them and then replying with them.

So, congratulations are in order now also for your utterly petty, sophomorically sarcastic, and incorrect snipe. Well done.

As far as I can think it through, your first paragraph seems reasonable and good added information to my own remark..(Though when I said 'by certain means', that of course encompasses whatever means do fail, such as the constructive ones you mention.)

As for the second paragraph, I don't know what you mean by an ordinal that is not well-founded. Any ordinal is well-founded by the membership relation, of course.

For one, I was correct, and you would long have realized it if you weren't so contrarian. For two, your little experiment made you look like you were ignoring your interlocutors.

I've read of some scientific papers recently that indicate that children have at least a preliminary understanding of number from a very young age. This leads to the hypothesis that a sense of number is inborn, instinctual, just as our ability to learn and use language is. — T Clark

The thing about notions like ‘inborn’ and ‘instinctual’ is that they don’t differentiate between whole hog pre-formed contents and a capacity to learn to construct in stages a complex activity. Language and number I think are good examples of phenomena that can be understood in either way. Chomsky and Fodor belong to the ‘whole hog innate content’ group, believing inborn semantic as well as syntactic contents.

The thing about number and calculation is that they are not one simple thing , but mean different things in different cultural eras. Even when we begin from an agreed upon definition of number , observing the performance of very young children doesn’t tell us how much perceptual constructive activity was necessary for that child to get to the point where concepts like object and multiplicity made sense for them.

Of course, you persist petulantly. No, you were not correct. You gave the impression that I had just learned the quote function, which is not correct.

And you were incorrect about LEM, as I amply explained. And It's not contrarianism simply to explain that the connection you claimed between the LEM and paradox is incorrect and is a basic misunderstanding of logic.

And, yes, as I mentioned, I knew that not using the quote function had disadvantages, including the one you just mentioned.

This incident with you, in line with ones with other posters with misconceptions about logic and mathematics (not in philosophy, but in the mere rote, technical facts) confirms my thought that on forums such as this, it is virtually impossible to post corrections and explanations without there being posters who will bristle personally about that.

The thing about notions like ‘inborn’ and ‘instinctual’ is that they don’t differentiate between whole hog pre-formed contents and a capacity to learn to construct in stages a complex activity. Language and number I think are good examples of phenomena that can be understood in either way. Chomsky and Fodor belong to the ‘whole hog innate content’ group, believing inborn semantic as well as syntactic contents. — Joshs

My level of understanding comes from reading summaries of a few scientific papers and then "The Language Instinct." My intent was not to make a strong case for any differing views, but just to point it out as an interesting sidelight.

You sure bristle about it quite a lot yourself.

What's the meaning of the LEM, according to you?

I don't bristle against being corrected on matters of logic.

I don't know what scope you have in mind by 'meaning' but I take the LEM in its utterly ordinary sense:

Syntactically: P v ~P

Semantically: Every sentence in the language is either true in the model or it is false in the model (where 'or' is the inclusive or; while the 'but not both' clause for exclusive or is demanded by the law of non-contradiction: ~(P & ~P)).

They only "hold sway" in a few academic circles that are irrelevant to anything. — Olivier5

I think the rejection of 'innate ideas' and mathematical intuitions is characteristic of a much wider circle than lumpen materialism. The 'other scholars' that the quote refers to are not necessarily eliminative materialists. In the IEP article on the 'indispensability' argument for mathematics - ' [Rationalist philosophers] claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.' None of those referred to in that article are eliminative materialists.

This leads to the hypothesis that a sense of number is inborn, instinctual, just as our ability to learn and use language is. — T Clark

Which empiricism generally resists, on the grounds that humans are born 'tabula rasa', a blank slate, on which ideas are inscribed by experience.

Which empiricism generally resists, on the grounds that humans are born 'tabula rasa', a blank slate, on which ideas are inscribed by experience. — Wayfarer

Well, there’s metaphysical , or ‘naive’ realism , which tends to be linked with Enlightenment associationism, and then there’s representational realism , which is often associated with neo-Kantianism. The former was consistent with behavioristic approaches in psychology , while the latter is compatible with cognitive science. Embodied versions of cog sci reject tabula rasa in favor of a cognizing subject already pre-situated by virtue of both learned schemata and inborn predispositions. And yet it considers itself a fully naturalistic empiricism.